T-junctions in spline surfaces Ke ¸ stutis Karˇ ciauskas Vilnius University and Daniele Panozzo New York University and J¨ org Peters University of Florida T-junctions occur where surface strips start or terminate. This paper de- velops a new way to create smooth piecewise polynomial free-form spline surfaces from quad-meshes that include T-junctions. All mesh nodes are interpreted as control points of GT-splines, i.e. geometrically smoothly joined piecewise polynomials. GT-splines are akin to and compatible with B-splines and cover simple T-junctions by two polynomial pieces of degree bi-4 and more complex ones by four such patches. They complement multi- sided surface constructions in generating free-form surfaces with adaptive layout. Since GT-splines do not require a global coordination of knot intervals, GT-constructions are easy to deploy and can provide smooth surfaces with T-junctions where T-splines can not have a smooth parameterization. GT- constructions display a uniform highlight line distribution on input meshes where alternatives, such as Catmull-Clark subdivision, exhibit oscillations. Categories and Subject Descriptors: Computing Methodologies [Computer Graphics]: Parametric Curve and Surface Models—spline surfaces General Terms: T-junctions, Surfaces Additional Key Words and Phrases: T-junctions, spline, surface, smooth- ness, highlight line distribution ACM Reference Format: acmformat This work was supported in part by NSF grant CCF-1117695, NSF CAREER award 1652515, NIH grant R01 LM011300-01 and DARPA TRADES HR00111720031 Authors’ addresses: Vilnius University, LT-2006, Vilnius, Lithuania; New York University, NY, USA; University of Florida, Gainesville FL 32611- 6120, USA, [email protected]fl.edu, tel/fax 352.392.1200/1220 Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permis- sion and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or [email protected]. c YYYY ACM 0730-0301/YYYY/17-ARTXXX $10.00 DOI 10.1145/XXXXXXX.YYYYYYY http://doi.acm.org/10.1145/XXXXXXX.YYYYYYY 1. INTRODUCTION Where strips of surface patches are forced together, it is natural to terminate some; and where strips are stretched wide, it is natural to spawn additional strips to keep the size and aspect ratio of the patches within bounds. Stopping or initiating surface strips leads to T-junctions where two finer surface pieces meet one coarser piece The simplest T-junction-configuration, a ˙ T -net (pronounced T1- net), is shown in Fig. 1a: a nominally pentagonal face with exactly one vertex of valence 3 is surrounded by quadrilateral facets. In isolation such transitions are easily modeled by smooth hierarchi- cal splines. But as part of a larger model, their knot-intervals need to be globally coordinated. That coordination is cumbersome. The small quad-mesh in Fig. 2 shows that it may even be impossible. (a) ˙ T -net layout (b) convex ˙ T -net Fig. 1. A control net with a single isolated T-junction. T-junctions allow introducing geometry of higher detail, or to merge two separately-developed spline surfaces T-junctions also prominently arise when replacing the complex and global con- straints of strict quad-meshing [Bommes et al. 2012; Vaxman et al. 2016] by T-meshes, based on triangle meshes [Li et al. 2006; Lai et al. 2008], curvature directions [Alliez et al. 2003; Marinov and Kobbelt 2004], directional fields [Myles et al. 2010; Myles et al. 2014a; Pietroni et al. 2016], optimized for planarity [Zadravec et al. 2010; Peng and Wonka 2013] or extracted from local parametriza- tions [Ray et al. 2006; Jakob et al. 2015]. T-junctions and hierarchical splines. One approach to incor- porating T-junctions is hierarchical splines (see e.g. [Kraft 1998; Sederberg et al. 2003; Giannelli et al. 2012; Dokken et al. 2013; Kang et al. 2015]). Hierarchical splines require that all surface pieces share a single uv-parameterization: for any choice of v, the u-knot intervals must add to the same number; and for any choice of u, the v-intervals must add to one fixed number. This restriction on the knot sums is natural when refining a single patch. But when the input is a given quadrangulation the knot intervals have to be ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
Transcript
T-junctions in spline surfaces
Kestutis Karciauskas
Vilnius University
and
Daniele Panozzo
New York University
and
Jorg Peters
University of Florida
T-junctions occur where surface strips start or terminate. This paper de-
velops a new way to create smooth piecewise polynomial free-form spline
surfaces from quad-meshes that include T-junctions. All mesh nodes are
interpreted as control points of GT-splines, i.e. geometrically smoothly
joined piecewise polynomials. GT-splines are akin to and compatible with
B-splines and cover simple T-junctions by two polynomial pieces of degree
bi-4 and more complex ones by four such patches. They complement multi-
sided surface constructions in generating free-form surfaces with adaptive
layout.
Since GT-splines do not require a global coordination of knot intervals,
GT-constructions are easy to deploy and can provide smooth surfaces with
T-junctions where T-splines can not have a smooth parameterization. GT-
constructions display a uniform highlight line distribution on input meshes
where alternatives, such as Catmull-Clark subdivision, exhibit oscillations.
Categories and Subject Descriptors: Computing Methodologies [Computer
Graphics]: Parametric Curve and Surface Models—spline surfaces
General Terms: T-junctions, Surfaces
Additional Key Words and Phrases: T-junctions, spline, surface, smooth-
ness, highlight line distribution
ACM Reference Format:
acmformat
This work was supported in part by NSF grant CCF-1117695, NSF
CAREER award 1652515, NIH grant R01 LM011300-01 and DARPA
TRADES HR00111720031
Authors’ addresses: Vilnius University, LT-2006, Vilnius, Lithuania; New
York University, NY, USA; University of Florida, Gainesville FL 32611-
Where strips of surface patches are forced together, it is natural toterminate some; and where strips are stretched wide, it is naturalto spawn additional strips to keep the size and aspect ratio of thepatches within bounds. Stopping or initiating surface strips leads toT-junctions where two finer surface pieces meet one coarser piece
The simplest T-junction-configuration, a T -net (pronounced T1-net), is shown in Fig. 1a: a nominally pentagonal face with exactlyone vertex of valence 3 is surrounded by quadrilateral facets. Inisolation such transitions are easily modeled by smooth hierarchi-cal splines. But as part of a larger model, their knot-intervals needto be globally coordinated. That coordination is cumbersome. Thesmall quad-mesh in Fig. 2 shows that it may even be impossible.
(a) T -net layout (b) convex T -net
Fig. 1. A control net with a single isolated T-junction.
T-junctions allow introducing geometry of higher detail, or tomerge two separately-developed spline surfaces T-junctions alsoprominently arise when replacing the complex and global con-straints of strict quad-meshing [Bommes et al. 2012; Vaxman et al.2016] by T-meshes, based on triangle meshes [Li et al. 2006; Laiet al. 2008], curvature directions [Alliez et al. 2003; Marinov andKobbelt 2004], directional fields [Myles et al. 2010; Myles et al.2014a; Pietroni et al. 2016], optimized for planarity [Zadravec et al.2010; Peng and Wonka 2013] or extracted from local parametriza-tions [Ray et al. 2006; Jakob et al. 2015].
T-junctions and hierarchical splines. One approach to incor-porating T-junctions is hierarchical splines (see e.g. [Kraft 1998;Sederberg et al. 2003; Giannelli et al. 2012; Dokken et al. 2013;Kang et al. 2015]). Hierarchical splines require that all surfacepieces share a single uv-parameterization: for any choice of v, theu-knot intervals must add to the same number; and for any choiceof u, the v-intervals must add to one fixed number. This restrictionon the knot sums is natural when refining a single patch. But whenthe input is a given quadrangulation the knot intervals have to be
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
2 • K. Karciauskas et al.
Fig. 2. T-splines require that the sum of knot intervals on opposing edges
of any face must be equal (Rule 1 of [Sederberg et al. 2003]). This forces
the width of the horizontal knot intervals of the grey helical strip to be zero,
preventing a smooth T-spline parameterization. (cf. Fig. 13 for a smooth
GT-spline surface.)
assigned and hence coordinated. Joining many pieces can then be-come cumbersome since the local knot intervals have to globallyadd up to matching sums. Where the mesh is not regular, the localconstruction in [Wang et al. 2011] is therefore only C0 despite gen-erating one order of magnitude more patches than quads. [Seder-berg et al. 2008] additionally points to shape deficiencies whereT-splines with dissimilar knot spacings are merged and proposesspecial splines of much higher degree. The global construction of[Li et al. 2006] introduces additional T-joints and new extraordi-nary points that are not motivated by geometry but solely by enforc-ing knot interval constraints (Pre-dating [Li et al. 2012], the con-struction does not guarantee ’analysis-suitable’ knot-distributions.)Akin to [Li et al. 2006, Fig.5], the example in Fig. 2 demonstratesthat, without modifying the quad-mesh, global coordination is notalways possible without loosing smoothness and even continuity ofthe parameterization. In the example, each red strip forms a braceletthat is half as wide when it comes back to meet up with the start-ing edge. Since Rule 1 of any T-spline construction according to[Sederberg et al. 2003] mandates that the horizontal knot intervalof the red strip be the same where the single-wide edge meets thedouble-wide edge, the horizontal knot interval of the grey helicalstrip of patches must be zero. Since, in the example, three consec-utive grey horizontal knot intervals are zero, the degree bi-3 splineparameterization is formally C−1. That is, the most basic prop-erty of T-splines prevents a smooth parameterization for a classof patch layouts that could be hidden in any large scale quad ar-rangement. The bracelet implies that joining spline surfaces with asmooth T-spline parameterization is not always feasible since this,too, requires making equal the knot interval sums. The slightlylarger mesh of Fig. 18a demonstrates a more complex incompati-bility with any assignment of knot-intervals for smooth T-splines.In summary, while hierarchical splines are naturally suited for in-troducing T-junctions in quad meshes, they are not naturally suitedfor generating surfaces from quad meshes with T-junctions.
T-junctions and Catmull-Clark subdivision. A strictly localconstruction is provided by Catmull-Clark subdivision [Catmulland Clark 1978]. Here the underlying model is splines with uni-form knot spacing and local support. Therefore the parameteriza-tions need not be globally coordinated. However, as Fig. 3a demon-
strates for the convex input T -net of Fig. 1a, the resulting surfacescan be of poor quality: the silhouette dips and rises and the high-light lines oscillate near the T-junction. We note that the oscilla-tions already manifest themselves in the first two subdivision stepsand hence rule out Catmull-Clark mesh refinement even as a pre-processor for turning a T-junction into a pair of isolated verticesof valence three and five. (We verified these ‘first step artifacts’[Augsdorfer et al. 2011] by replacing, in a separate computation,the red limit surface in Fig. 3a by a high-quality surface construc-tion.)
T-junctions and GT-splines. This paper develops a new lo-cal construction, a geometric approach to T-junctions. This GT-
(a) Catmull-Clark subdivision
(b) this paper: GT-splines
Fig. 3. Surfaces generated from the convex input T -net of Fig. 1b. The red
regions in (a) represent an infinite sequence of bi-3 patches covering the 5-
sided and 3-sided extraordinary Catmull-Clark neighborhoods arising from
a T -net. The red region in (b) consists of two bi-4 patches by which the
GT-construction covers the T -net. Catmull-Clark subdivision produces, in
the first two steps, a flattened silhouette and a correspondingly non-uniform
highlight line distribution (right).
construction does not require global knot interval coordination andyields better shape than Catmull-Clark subdivision (see Fig. 3a vs3b). The GT-construction is based on reparameterization, the natu-ral technique for transitioning between unequal parameterizationson opposite sides of a T-junction. Consequently the construction
leverages the framework of geometric continuity. For T -nets, theresulting surfaces
—consist of a frame of bi-3 (bi-cubic) patches filled by two patchesof degree bi-4 (this ‘cap’ is red in Fig. 1b).
—The bi-4 cap is internally smooth and joins the bi-cubic patcheswith tangent continuity (G1)
—The bi-4 cap yields good highlight line distributions on all of alarge number of challenging input meshes.
—The Appendix provides simple explicit formulas for all relevantBernstein Bezier coefficients of the GT-construction in terms ofthe local T -net.
—GT-splines complement few-piece polynomial constructionssuch as [Karciauskas et al. 2016] covering extraordinary pointsto model smooth free-form surfaces of maximal degree bi-4.
We also present two variants of the GT-construction, to generatesmooth caps for multiple T-junctions within one facet whose ex-tensions cross or are parallel.Overview. Section 2 reviews basic concepts of the construction ofsmooth surfaces. Section 3 defines a bi-3 frame of patches that tran-sitions to the surrounding surface. Section 4 describes the G1 bi-4
GT-construction of the cap for T -nets. Section 5 and Section 6 de-velop caps for cases of adjacent T-junctions: for two T-junctionsopposite one and for two T-junctions with crossing directions. Sec-tion 7 compares the constructions for challenging input data, ex-plains the choices taken along the way and lists the limitations.Section 8 shows how GT-splines collaborate with algorithms in theliterature to smoothly cover multi-sided neighborhoods by an over-all smooth bi-4 surface.
2. DEFINITIONS AND SETUP
We will construct a T-junction surface by averaging alternative in-terpretations of the mesh points as regular control points (see Fig. 5
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
T-junctions in spline surfaces • 3
and Section 3). This yields a frame of bi-3 polynomial patches tobe filled by a cap consisting of polynomial patches of degree bi-4.The following definitions make this approach precise.
The GT-splines are a collection of tensor-product patches inBernstein-Bezier form (BB-form; see e.g. [Farin 1988]):
f(u, v) :=d
∑
i=0
d∑
j=0
fijBdi (u)B
dj (v) , (u, v) ∈ [0..1]2,
where Bdk(t) :=
(
d
k
)
(1 − t)d−ktk are the BB polynomials of de-gree d and fij are the BB-coefficients. Adjacent patches join withGk continuity if their kth-order jets (one-sided Taylor expansion)match along their common boundary after a change of variables ρ.This characterization is equivalent to formulations of Ck continuityof manifolds in terms of charts, see e.g. [Peters 2002]. We use the
succinct characterization that two surface pieces f and f sharing aboundary curve e join G1 if there is a suitably oriented and non-
singular reparameterization ρ : R2 → R2 so that the jets ∂k f and
∂k(f ◦ ρ) agree along e for k = 0, 1. Although ρ is just a changeof variables, its choice is crucial for the properties of the resultingsurface. Throughout, we will choose e to correspond to the patchparameters (u, 0 = v). Then the relevant Taylor expansion of thereparameterization ρ with respect to v is ρ := (u+ b(u)v, a(u)v)and the chain rule of differentiation yields the G1 constraints
∂v f(u, 0)− a(u)∂vf(u, 0)− b(u)∂uf(u, 0) = 0. (1)
If f and f are polynomials then a and b are rational functions whose
degree is bounded in terms of the degree of f and f [Peters 1991].
(a) Extended T -net (b) Bi-3 neighborhood
Fig. 4. An isolated T-junction in an extended T -net (a) provides (b) a bi-3
neighborhood (solid,for context only) and a C2-prolongation in BB-form
(inner green mesh) that is the only part used for the GT-construction.
C1 continuity of the splines over non-uniform knot sequencescan be recast as G1 continuity of patches defined over unit do-mains (see e.g. [Karciauskas and Peters 2011]). For example, if f
and f are consecutive curve segments originally associated withintervals [−1, 0] and [0, 1
2] of a C1 spline with knot sequence
{. . . ,−1, 0, 1
2, . . .} then both f and f can be newly defined, each
on the interval [0, 1], and they then join as β∂uf(1) = ∂u f(0) withβ := 1/2. When we want to point out that surface patches are, inone variable, related by the identity and, in the other, by C1 con-tinuity over non-uniform knot sequence, we refer to the transitionas: C1 with parameter β.
Our main construction focusses on T -nets that consist, as shownin Fig. 1, of quadrilaterals and one nominally five-sided facet. For
context and exposition, we can extend the T -net by one layer ofquadrilaterals (see Fig. 4a). This allows applying, away from the
five-sided facet, the well-known bi-cubic (bi-3) B-spline to BB-form conversion rules (see e.g. [Farin 1988]). The resulting C2 bi-3
spline neighborhood is colored brown in Fig. 4b. The smaller T -net( Fig. 1 ) provides a tensor-border of degree 3 and depth 2, the C2-prolongation of second-order Hermite data shown as a green net of
BB-coefficients in Fig. 4b. Given the tensor-border, the T -net in-terior of 4 + 4+ 5+ 5 control points (see the stencils in Fig. 5d,f)provides all the information for the GT-construction. All formulas
(stencils) of the bi-4 cap in terms of the T -net interior are providedin the Appendix.
ql
(a) ql
qr
(b) qr
l1
l0
l1m
r1
r0
r1m
(c) Interpretation as BB-patches of degree bi-3, overlaid in (e)
55 11
32 88
8 22
(d) ✷ × 172
bdbk
tdtk
(e) frame
22 4646
184184 88
46 2424 11
(f) ◦ × 1576
Fig. 5. Constructing the frame. The regular left (a) and right (b) control
nets obtained by re-connecting the nodes of the T -net define six bi-3 patches
each. (e) Completion of the frame. (d,f) Stencils of the points qb,−1✷ = q
b,1✷
marked ✷ and qt,−1◦ = q
t,1◦ marked by ◦. in (e).
3. CONSTRUCTION OF A BI-3 FRAME OF
PATCHES FOR CAPPING A T -NET
The surface corresponding to a T -net will consist of a frame ofbi-3 patches and a central cap consisting of two bi-4 patches. Thissection builds the frame. For i = −1, 0, 1 the frame has left patchesql,i, right patches qr,i, (Fig. 5c) and for j ∈ {−1, 1} top patchesqt,j and bottom patches qb,j (Fig. 5e). The ribbon is derived by
re-connecting the nodes of the T -net to form two regular nets, ql
from the left (see Fig. 5a) and qr from the right (see Fig. 5b).
We now interpret ql and q
r as bi-3 B-spline control nets andconvert them to BB-form. As illustrated in Fig. 5c the so-derived
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
4 • K. Karciauskas et al.
patches agree – except for the lower boundary curves of the toppatches and the upper boundary of the bottom patches. The bottomcurves overlap only at their endpoints, marked as a magenta and acyan box in Fig. 5c. A new common BB-control point, also markedas a blue box in Fig. 5e, is chosen to be the average of these two can-didates: Fig. 5d displays the explicit stencil for the common point
q✷ := (qb,−1✷ +q
b,1✷ )/2 obtained by this averaging. The two direct
neighbors of q✷ are chosen to make the combined boundary curveC2. (The blue disks are defined by the C1-prolongation of ql,−1
and qr,−1). The top patches are subdivided at their midpoint andthe resulting overlap is then treated like that of the bottom: Fig. 5fdisplays the explicit stencil for the common point q◦ obtained bythis averaging Although this split into qt,−1 and qt,1 serves onlyto accommodate the lower boundary of the top patches, extensiveexperiments show this split to be critical for achieving good shape(see e.g. Fig. 12). The resulting frame of bi-3 patches is C2 exceptalong the four hv-curves, the red curves in Fig. 6c between the hor-izontal and the vertical strips of the frame: ql,−1 to qb,−1, ql,1 toqt,−1, qr,−1 to qb,1, qr,1 to qt,1. Across the hv-curves the continu-ity is C1. Since the construction did not change the BB-coefficientsderived from q
l and qr that match the tensor-border (Fig. 4b), the
frame joins C2 with the splines surrounding it.
l
t
b
v
u
v
u
v
u
(a) C1-prolongations
p2p1
(b) central bi-4 patches
(c) final cap
Fig. 6. Construction of the bi-4 cap. (a) the mismatch of the C1-
prolongations is resolved by reparameterizing them. (b) Interior coefficients
minimize the distance to bi-cubics. (c) final layout: the bi-3 patches are C1-
connected across the (red) hv-curves.
4. CENTRAL T -NET CAP CONSTRUCTION
To complete the surface, we construct a central cap (red in Fig. 6) ofdegree bi-4 that fills the frame so that all transitions are at least G1.The bi-4 cap consists of two patches pl, pr; see Fig. 6c. Since theconstruction of pr mirrors that of pl, we discuss only pl. Fig. 6ashows the C1-prolongations t of qt,−1 and b of qb,−1 in black andl of ql,0 in green. While b is consistent with l, the prolongations tand l are inconsistent (due to the split of the top patch when con-structing the frame). Since we reparameterize l linearly (to mini-mize the final patch degree) to match t, we also need to reparam-eterize b after all. Together, the choice of parameterizations in Eq.
(1) are
a(u) := b(u) :=left(l) : 1− u
20
top(t) : 1 (1− u)ubottom(b) : 1 − 1
2(1− u)u.
(2)
The interior BB-coefficients (circles in Fig. 6b) of the bi-4 cap aredetermined so that columns of BB-coefficients form degree-raisedcurves of true degree 3. By construction, see Fig. 6c, the red bi-4 cap is internally C1 and joins with G1-continuity to the greenframe.
5. CAPS FOR PARALLEL T-JUNCTIONS
Configurations with multiple T-junctions can in principle be locally
re-meshed to separate them into isolated T -nets. For completeness,and to compare what surface quality can be achieved, we investi-gate configurations where two T-junctions face another, as shown inFig. 7b. We call the configuration a
...T -net (pronounced T3-net). It
has one nominally 7-sided face. Such T-junctions can arise, for ex-ample, from configuration Fig. 7a by removing the two red edges ofa triangle attached to a point of valence 5. (Asymmetrically remov-ing one yields a mesh with one T-junction as in Fig. 7c). ApplyingCatmull-Clark subdivision to
...T -junctions leads to poor surfaces.
As for T -nets, first a bi-3 frame is constructed. Fig. 7e,f providethe stencils for the points marked ✷ and ◦. Except across the redhv-curves between the horizontal and the vertical strips, the frame
is C2. In the spirit of the bi-4 T -net construction, a bi-4 cap
(a) pre-...T -mesh (b)
...T -net (c) re-mesh
(d) frame
7 14
32 88
8 22
(e) ✷ × 172
4 11
16 44
3 3
(f) ◦ × 136
Fig. 7. Construction of the frame for a...T -junction.
for the...T -net is constructed by subdividing, in the ratio shown in
Fig. 8a, the C1-prolongation of the frame from the top; and thenevenly splitting the middle prolongation of the bottom. Both thetop and bottom prolongations are C1-connected (in the horizontaldirection) with the same continuity parameters from left to right:β = 1
2, β = 1, β = 2. This implies a continuity parameter of
β = 2
3across the top hv-curves. For the prolongation l of ql,0
(see Fig. 8b) to match the split prolongation of the top, the data arereparameterized according to
a(u) := b(u) :=left(l) : 1− u/3 0top(left) : 1 1
2(1− u)u
bottom(left) : 1 − 1
3(1− u)u.
(3)
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
T-junctions in spline surfaces • 5
(For the right top and bottom reparameterizations, b(u) is negated.)The remaining (circled) BB-coefficients in Fig. 8 are chosen tomake their columns have actual degree 3.
: 12
: 11
(a) splitting the pro-
longation
v
u
v
u
v
u
v
u
v
u
(b) reparameteri-
zation
(c) bi-4 cap
Fig. 8. bi-4 cap for a...T -net.
6. CAPS FOR TWO T-JUNCTIONS IN CROSSING
DIRECTIONS
We also investigate configurations with two T-junctions as shown
in Fig. 9a. We call the configuration a T -net (pronounced T2-net).It has one nominally 6-sided face. Note that such configurations areexplicitly excluded in dyadic T-meshes [Kovacs et al. 2015].
(a) T -net (b) frame
(c) reparameterizations (d) bi-4 cap (red)
Fig. 9. Bi-4 cap for a T -configuration. The red axes in (c) indicate the
v-parameter.
Capping T -nets with good surface quality is similar but morechallenging than the earlier constructions. As before, B-spline toBezier conversion yields the green Bezier control points in Fig. 9b,now with both the top and the right side patches subdivided.
The corner points (blue boxes in Fig. 9b) and the middle, bluecircle coefficients of innermost (blue) boundary curves of the frameare determined only in the last step of the construction. The di-rect neighbors of the corner points are chosen so that adjacent bi-3patches of the frame connect C1 (with ratios 1,1 lower left, 1/2,1/2upper right and 1/2,1 otherwise); the direct neighbors of the middlepoints are determined so that each boundary curve is internally C2.
The C1-prolongations are not compatible at the corners (seeFig. 9c). To make them compatible they are re-parameterized with
a(u) := b(u) :=top− left : 1− u
4
1
2(1− u)u
top− right : 3
4− u
4− 1
2(1− u)u
bottom− left : 1− u4
− 1
4(1− u)u
bottom− right : 3
4− u
4
1
4(1− u)u.
This list of the reparameterizations is complete due to the (com-binatorially) diagonal symmetry (see the local coordinate systemsin Fig. 9c). The reparameterized tensor-border is of degree 4 anddepth 1 and ensures G1 continuity of the central cap with theframe. Choosing to join C2 the four 3 × 3 groups of interior BB-coefficients leaves free one group shown as red disks. Finally, weminimize, over all 11 bi-3 patches of the frame and the 4 bi-4patches of the central cap, the functional F3 where
Fκf :=
∫ 1
0
∫ 1
0
∑
i+j=κ,i,j≥0
κ!
i!j!(∂i
s∂jt f(s, t))
2dsdt.
We minimize the sum with respect to 17 unknown coefficients: 4corner (blue box), 4 mid-edge (blue circle) and 9 inner ones (reddisks in Fig. 9d). In the implementation, these 17 coefficients enter
as affine combinations of T -net points with pre-computed coeffi-cients.
The choice F3 is the result of testing a series of input meshes in-cluding the challenging elliptic configuration in Fig. 10. Fig. 11confirms that this choice also works well for a wave-like inputmesh.
(a) T -net (b) BB-nets (c) default F3
(d) F2 (e) F4 (f) F5
Fig. 10. The surfaces obtained by minimizing functionals. (a) convex T -
net. (b) BB-coefficients of the frame (green) and the central bi-4 cap (red).
(c-f) Highlight lines.
(a) T -net (b) frame and cap (c) highlight lines
Fig. 11. A T -net (F3) surface for a non-convex input mesh.
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6 • K. Karciauskas et al.
7. DISCUSSION AND COMPARISON
7.1 The frame construction: to split or not to split
Note the flaws in the highlight line distribution of the surface inFig. 12b. For the same input mesh Fig. 1b, the highlight line distri-bution of Fig. 12b is much worse than that of the GT-constructionFig. 3b. Despite being formally smooth, surfaces generated bynot subdividing the frame patch opposite the T-junction have poorhighlight lines. Evidently splitting, though not required by formalsmoothness contraints, improves shape quality.
(a) cap layout (b) highlight lines
Fig. 12. An alternative construction that does not split the patch opposite
to the T-junction leads to a poor highlight line distribution.
7.2 Comparing GT-splines to T-splines
In general a comparison to hierarchical splines does not make sensesince we proved at the outset that not all meshes with T-junctionsadmit smooth T-splines. For one such configuration, the braceletmesh of Fig. 2 (that does not admit a C1 T-spline), Fig. 13 demon-strates that applying GT-splines yields a bi-4 surface with an excel-lent highlight line distribution.
Fig. 13. The GT-spline surface for the T-mesh of Fig. 2 that does not admit
a C1 T-spline. The bi-4 cap in red; the right image shows highlight lines.
(a) T -net (b) highlight lines (c) mean curvature
Fig. 14. Two regular meshes are merged with a T-junction.
Fig. 14 shows the case of two regular meshes of different quad-patch count joined via a T-junction. T-splines require forming acommon parameter domain, that, while easy for simple meshes, isimpossible for more complex meshes such as Fig. 18. The result ofdirectly applying a GT-spline is displayed in Fig. 14b,c. To estab-lish an upper bound on the quality of the highlight line distribution,Fig. 15 compares GT-splines to T-splines in the form of standard bi-cubic tensor-product splines. The top row of Fig. 15 shows in order
(a) the T -net input to the GT-construction, (b) the geometricallyidentical T-mesh, with all knot intervals are 1 except for 0.5 on thethick edges; and (c) the mesh resulting from splitting the cyan knot
Fig. 15. Comparisons to T-splines and tensor-product splines. Since, in
these examples, the T- and B-spline surfaces coincide, we refer to them as
(T-)spline surfaces.
intervals yielding new (cyan disks) and moved (cyan circles) con-trol points. Since the T-spline surface of input mesh (b) equals thenon-uniform C2 tensor-product B-spline surface of input mesh (c),it suffices to discuss the case (c) in the following. For a concretecomparison in this setting, the mesh in Fig. 15d is the geometricinput both for the GT-spline and for the T-spline, whereas Fig. 15eis the tensor-product B-spline input mesh. As explained above, theknot-intervals are chosen so that the B-spline surface equals the T-spline surface. Fig. 15f,g show the highlight line distribution on thesurfaces.
In Fig. 15h two regular meshes are connected by a whole ring ofT-junctions. This is the (geometric) input mesh both for the GT-spline and for the T-spline, but the knot-intervals differ: for theGT-spline they can all be chosen equal, while the horizontal T-spline knot intervals of bottom mesh are 0.5 when those of the topmesh are 1. Fig. 15i shows the tensor-product mesh yielding theC2 surface Fig. 15j that coincides with T-spline surface from meshFig. 15h. Fig. 15k shows the GT-spline surface obtained directlyfrom Fig. 15h. The highlight lines of the GT-construction are verysimilar to those of the (T-)spline construction even though the GT-spline is placed at a disadvantage by capping T-junctions while the(T-)spline can take advantage of a tensor-product mesh.
Given the similarity in shape, it is important to recall the essen-tial difference between T-splines and GT-splines. The simple knotintervals used locally above can lead to invalid knot-intervals whenconsidering larger meshes: T-splines require a global coordinationof knot sequences. Such global coordination is not always possibleor may require complex re-meshing. The GT-construction is local,sidestepping the need for global coordination, and producing sur-faces of comparable quality.
7.3 Separation and Re-meshing
The GT-constructions in Section 4, 5 and 6 follow a commonpattern, of adjusting the bi-3 frame and then forming a central
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
T-junctions in spline surfaces • 7
cap of degree bi-4. The minimal submesh required for the GT-constructions is called ‘net’. Only the outer boundaries of the framemay have irregular nodes, where n 6= 4 quads meet, or T-junctions.Guaranteeing such separation may not be easy in general (see Limi-tations below) but often local adjustments can be made. While Sec-
tion 6 demonstrated that T -nets can yield bi-4 surfaces of goodquality, better surfaces are often obtained by locally re-connectingthe mesh points to isolate the T-junctions as in Fig. 16c. Reconnect-ing with an even larger footprint for a more symmetric reconnectionas in Fig. 16d can further improve the highlight line distribution, atthe cost of higher construction complexity.
(a) T -net (b) mesh and bi-4 cap
(c) asymmetric re-mesh (d) symmetric re-mesh
Fig. 16. Effect of re-meshing on the resulting bi-4 cap.
7.4 Limitations
Quadrangulations can contain closely packed T-junctions and ir-regular points. As our initial example demonstrated, Catmull-Clarkrefinement is not a good way to separate T-junctions: not only doessubdivision increase the number of patches, but, more importantly,it can negatively impact the surface quality. T-mesh subdivision[Kovacs et al. 2015] can also start with irregular points adjacent toT-junctions but does not separate T-junctions from irregular points.Therefore, it too cannot be used for pre-processing.
Although re-meshing can reduce many configurations to thethree standard T-nets, GT-constructions are not expected to workwith arbitrary T-junction distributions. Here, the quad-meshing al-gorithm or the designer have to enforce some discipline, already toobtain good shape. Many-sided facets with T-junctions as generatedby [Alliez et al. 2003] or motorcycle graphs [Eppstein et al. 2008;Myles et al. 2014b] are outside the scope of GT-constructions.
In many cases, our approach can allow for tighter packing of T-junctions (for example as in Fig. 15,bottom row and Fig. 18) andirregularities. However, this paper does not attempt to provide a setof recipes for arbitrarily complex T-junctions. On one hand, local
re-connection can often reduce the situation to a collection of T -net, T -nets or
...T -nets, but a principled prescription for such quad-
re-meshing is outside the scope. On the other hand, our experimentswith complex configurations show that keeping the complexity ofcapping T-junctions to a minimum results in better shape.
(a) input mesh (b) bi-3 green, bi-4 else
(c) highlight lines (d) mean curvature
Fig. 17. Mesh and bi-4 surface combining a T-junction with irregular re-
gions of valence 3 and 5 where [Karciauskas et al. 2016] is applied.
(a) input mesh (b) highlight lines
(c) bi-3 gold, bi-4 (blue, purple, red) (d) mean curvature
Fig. 18. Mesh and G1 surface including horizontally paired T -nets (red)
and irregular neighborhoods (blue,purple) treated with [Karciauskas et al.
2016]. This mesh does not admit a globally consistent (non-zero) knot in-
terval assignment for smooth T-splines.
8. COLLABORATION WITH MULTI-SIDED CAPS
Fig. 17 demonstrates that sufficiently isolated caps for T-junctionsco-exist without problems with irregular vertices where the surfacecaps are also of degree bi-4 when we apply [Karciauskas et al.2016]. Again, tighter configurations are possible, but may reducesurface quality while increasing the complexity of implementation.
Fig. 18 presents another free-form design that challenges algo-rithms that require globally consistent knot intervals. As in Fig. 2,enforcing T-spline Rule 1 yields zero knot intervals, now also atextraordinary points.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
8 • K. Karciauskas et al.
(a) T -net (b) T -net (c)...T -net
Fig. 19. Distribution of the bi-degree 4 patches (red) for the three basic
configurations. The frames (green) are of degree bi-3.
9. CONCLUSION
The paper introduced a construction of surface caps for merging
and spreading feature lines via T-junctions. In the default T -case,the caps consist of two surface patches of degree bi-4 Fig. 19, oth-erwise of four bi-4 patches. The surrounding bi-3 patches are onlyperturbed where they join the central cap. Since the approach isbased on geometric continuity, it does not require non-local coordi-nation of knot intervals and is not restricted to graphs of functionsbut applies to general manifolds.
REFERENCES
Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Levy, and
Amir Vaxman, Marcel Campen, Olga Diamanti, Daniele Panozzo, David
Bommes, Klaus Hildebrandt, and Mirela Ben-Chen. 2016. Direc-
tional Field Synthesis, Design, and Processing. Computer Graphics Fo-
rum (2016). http://graphics.tudelft.nl/Publications-new/
2016/VCDPBHB16
Wenyan Wang, Yongjie Zhang, Michael A. Scott, and Thomas J. R. Hughes.
2011. Converting an unstructured quadrilateral mesh to a standard T-
spline surface. Comput Mech 48 (2011), 477498.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
T-junctions in spline surfaces • 9
Mirko Zadravec, Alexander Schiftner, and Johannes Wallner. 2010. De-
signing Quad-dominant Meshes with Planar Faces. Computer Graphics
Forum 29, 5 (2010), 1671–1679. http://www.geometrie.tugraz.
at/wallner/vfdesign.pdf Proc. Symp. Geometry Processing.
Appendix
Due to symmetry and C2 continuity of the bi-3 frame with the sur-
rounding surface, the T -net-construction is completely defined bythe following formulas for one (here the ‘left’) bi-4 patch. As illus-trated for two of the coefficients in Fig. 5d,f, each of the 5× 5 BB-
coefficient is a linear combination of the inner T -net nodes withtwo rows of four weights and two rows of five weights. In Table I,the stencil for each BB-coefficient, is scaled × 144 and placed intobrackets.
[
416
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5 216
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660
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8
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8
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.
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
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